A361355 Triangle read by rows: T(n,k) is the number of simple series-parallel matroids on [n] with rank k, 1 <= k <= n.

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 15, 1, 0, 0, 0, 0, 75, 1, 0, 0, 0, 0, 735, 280, 1, 0, 0, 0, 0, 0, 9345, 938, 1, 0, 0, 0, 0, 0, 76545, 77805, 2989, 1, 0, 0, 0, 0, 0, 0, 1865745, 536725, 9285, 1, 0, 0, 0, 0, 0, 0, 13835745, 27754650, 3334870, 28446, 1, 0

Offset: 1

Author

Andrew Howroyd, Mar 09 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0, 0;
  0, 1,  0;
  0, 0,  1,   0;
  0, 0, 15,   1,     0;
  0, 0,  0,  75,     1,     0;
  0, 0,  0, 735,   280,     1,    0;
  0, 0,  0,   0,  9345,   938,    1, 0;
  0, 0,  0,   0, 76545, 77805, 2989, 1, 0;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
• Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
• Nicholas Proudfoot, Yuan Xu, and Ben Young, On the enumeration of series-parallel matroids, arXiv:2406.04502 [math.CO], 2024.

Crossrefs

Row sums are A007834.

Cf. A140945, A359985, A361353.

Cf. A034941, A361282.

Programs

PARI

\\ B gives A359985 as e.g.f.
B(n)= {exp(x*(1+y) + y*intformal(serreverse(log(1 + x*y + O(x^n))/y + log(1 + x + O(x^n)) - x)))}
T(n) = {my(v=Vec(serlaplace(log(subst(B(n), x, log(1 + x + O(x*x^n)))/(1 + x))))); vector(#v, n, Vecrev(v[n]/y, n))}
{ my(A=T(9)); for(i=1, #A, print(A[i])) }

Formula

E.g.f.: A(x,y) = log(1 + B(x,y)) where B(x,y) is the e.g.f. of A361353.

E.g.f.: A(x,y) = log(B(log(1 + x), y)/(1 + x)) where B(x,y) is the e.g.f. of A359985.

T(2*n+1, n+1) = A034941(n).

T(2*n, n+1) = A361282(n).